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Question:

A uniform thin rod AB of length L has linear mass density μ(x)=a+bx/L, where x is measured from A. If the CM of the rod lies at a distance of (7/12)L from A, then a and b are related as

a=2b

3a=2b

2a=b

a=b

Solution:

CM of the rod of length l:
xcm = ∫0L μ(x)xdx / ∫0L μ(x)dx
= ∫0L (a+bx/L)xdx / ∫0L (a+bx/L)dx
= aL2/2 + bL2/3 / aL + bL/2
=> 7/12L = aL2/2 + bL2/3 / aL + bL/2
=> 7/4 = 3a + 2b / 2a + b
=> 14a + 7b = 12a + 8b
=> 2a = b